### Abstract

A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M =sup max Ξ_{k}(t) t≥0 1≤k≤N(t) to be finite. Here Ξ_{k}(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODE σ^{2}(x)/2 f″(x) + a(x)f′(x) = λ(x)(1 -k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and κ(X) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x) ∫ π(x, dy) (f(y) - f(x)) and the product λ(x)(1 -κ(x))f(x) where λ(x) and κ(x) are as before, μ(x) is the intensity of jumping at point x, and π(x, dy) is the distribution of the jump from x to y.

Original language | English (US) |
---|---|

Pages (from-to) | 307-332 |

Number of pages | 26 |

Journal | Journal of Applied Mathematics and Stochastic Analysis |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1997 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

### Cite this

*Journal of Applied Mathematics and Stochastic Analysis*,

*10*(4), 307-332. https://doi.org/10.1155/S1048953397000397

}

*Journal of Applied Mathematics and Stochastic Analysis*, vol. 10, no. 4, pp. 307-332. https://doi.org/10.1155/S1048953397000397

**Boundedness of one-dimensional branching Markov processes.** / Karpelevich, F. I.; Soukhov, Iouri M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Boundedness of one-dimensional branching Markov processes

AU - Karpelevich, F. I.

AU - Soukhov, Iouri M.

PY - 1997/1/1

Y1 - 1997/1/1

N2 - A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M =sup max Ξk(t) t≥0 1≤k≤N(t) to be finite. Here Ξk(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODE σ2(x)/2 f″(x) + a(x)f′(x) = λ(x)(1 -k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and κ(X) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x) ∫ π(x, dy) (f(y) - f(x)) and the product λ(x)(1 -κ(x))f(x) where λ(x) and κ(x) are as before, μ(x) is the intensity of jumping at point x, and π(x, dy) is the distribution of the jump from x to y.

AB - A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M =sup max Ξk(t) t≥0 1≤k≤N(t) to be finite. Here Ξk(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODE σ2(x)/2 f″(x) + a(x)f′(x) = λ(x)(1 -k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and κ(X) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x) ∫ π(x, dy) (f(y) - f(x)) and the product λ(x)(1 -κ(x))f(x) where λ(x) and κ(x) are as before, μ(x) is the intensity of jumping at point x, and π(x, dy) is the distribution of the jump from x to y.

UR - http://www.scopus.com/inward/record.url?scp=0000316668&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000316668&partnerID=8YFLogxK

U2 - 10.1155/S1048953397000397

DO - 10.1155/S1048953397000397

M3 - Article

VL - 10

SP - 307

EP - 332

JO - International Journal of Stochastic Analysis

JF - International Journal of Stochastic Analysis

SN - 2090-3332

IS - 4

ER -